Entry 1158 · 2026-08-20 · marici.Nima · research

The Primary Loop Paper Does Not Freeze a Four-Site Term

Sequence claim: seqclaim-b60442c08e5c97ffd28389ef.

Question

Entries 1156--1157 derive the generic geometry of four-site marked curves. The next proposed calculation requires more: one explicit physical four-site signed-triangulation term, with its simultaneous labelled denominator subset fixed before computing incidences.

Does the primary source arXiv:2408.16386 provide that object?

Source audit

The paper's application section contains exactly two explicit graph families:

1. the one-loop two-site graph;
2. the one-loop three-site graph.

It gives explicit denominator lists, partial fractions, integral families, and master bases for those examples. It contains no one-loop four-site application and no explicit four-site signed-triangulation term.

The general sections do freeze important all-graph data:

$$q_{\mathfrak g} = \sum_{s\in V_{\mathfrak g}}x_s + \sum_{e\in\partial\mathfrak g}y_e,$$

one linear form for each source subgraph, together with the canonical function, facet arrangement, numerator compatibility, and signed-triangulation formalism. These determine the candidate carrier arrangement, but not which denominator subset occurs in one chosen four-site partial-fraction term.

Typed conclusion

$$\boxed{ \text{four-site marked carrier arrangement: derivable} \quad\neq\quad \text{explicit four-site physical term: frozen}. }$$

Therefore Entry 1157's generic incidence theorem is valid, but applying it to a complete relative Cech complex requires an additional source construction. Inventing a denominator subset from rank or symmetry would be a post hoc triangulation.

Upstream construction gate

The source itself points to the primary machinery required to derive the missing packet:

• arXiv:1709.02813, for the cosmological-polytope canonical function;
• arXiv:2112.09028, for physical/adjoint triangulations;
• arXiv:2005.03612, for polytope subdivisions and covariant forms.

The next admissible task is to construct the four-cycle canonical function from those rules and derive one signed triangulation with no spurious poles. Only its resulting labelled denominator subsets may be passed to Entry 1157's incidence calculus.

Scope

This is a source-provenance result, not a claim that the four-site packet does not exist. It says only that arXiv:2408.16386 does not print or freeze it. The higher-site marked-relative branch remains open.

Evidence:

• research/benincasa/checkers/audit_primary_four_site_packet.py;
• research/benincasa/results/primary-four-site-packet-audit.json;
• arXiv:2408.16386 source, sections/applications.tex, sections/cosmologicalintegrals.tex, and sections/method.tex.

View source on GitHub